Lecture 1 - Internet Analysis Seminar

ENTIRE FUNCTIONS AND COMPLETENESS
PROBLEMS
A. POLTORATSKI
Lecture 1
A set of vectors in a Banach or Hilbert space is called complete if finite
linear combinations of its vectors are dense in the corresponding space
with respect to the standard topology generated by the norm. For
example, any basis, or any set containing a basis, is a complete set.
One of the first advanced examples of a complete set we see in the
standard analysis course is the set of monomials 1, x, x 2 , x 3 , ... in the
space C([a, b]) of all continuous functions on a closed interval with the
standard supremum norm. The Weierstrauss theorem tells us that the
set is complete in the space.
Completeness problems appear in many areas of analysis and its applications.
For instance, a function on a subset of the real line may
represent a wave and one may ask if it can be approximated, in a specified
sense, by linear combinations of specially selected functions, often
called harmonics. In modern terms, defining the criterion of approximation
amounts to defining the norm in the space. Verifying wether
any function in the space can be approximated by finite linear combinations
of harmonics is equivalent to proving that harmonics are complete
in the space. Problems of this kind gave name to a large and important
part of mathematics, Harmonic Analysis.
The role of harmonics in the above set-up can be played by a number of
different sets of functions, such as trigonometric functions, monomials,
complex exponentials, or specials functions such as Bessel functions,
Jacobi or Chebyshev polynomials or Airy functions originating from
Physics. The most common choices for the Banach space are L p spaces
and spaces of continuous or smooth functions with various norms.
Let us start by formulating some of the classical completeness problems
that will be discussed in this course.
1

2 A. POLTORATSKI
Bernstein’s problem
Let W : R → [1, ∞) be a continuous function satisfying x n = o(W (x))
for any n ∈ N, as x → ±∞. Denote by CW the space of all continuous
functions f on R such that f/W → 0 as x → ±∞ with the norm
||f||W = sup
R
|f|
. (0.1)
W
The famous weighted approximation problem posted by Sergei Bernstein
in 1924 [2] asks to describe the weights W such that polynomials
are dense in CW .
Bernstein’s problem can be viewed as a natural consequence of the
Weierstrauss theorem. If instead of C([a, b]) one tries to consider C(R)
with the same supremum norm, one immediately runs into an obstacle:
polynomials do not belong to such a space. Bernstein’s weighted sup
norm turns out to be the best way to remedy that situation. A slightly
more general version of Bernstein’s problem allows the weight W to be
semicontinuous and take infinite values. This extension allows one to
study polynomial approximation on arbitrary closed subsets of R.
Throughout the 20th century Bernstein’s problem was investigated by
many prominent analysts including N. Akhiezer, L. de Branges, L.
Carleson, T. Hall, P. Koosis, B. Levin, P. Malliavin, S. Mandelbrojt,
S. Mergelyan, H. Pollard and M. Riesz. This activity continues to
our day with more recent significant contributions by A. Bakan, M.
Benedicks, A. Borichev, P. Koosis, M. Sodin and P. Yuditski, among
others. Besides the natural beauty of the original question, such an extensive
interest towards Bernstein’s problem is generated by numerous
links with adjacent fields, including its close relation with the moment
problem.
Further information and references on the remarkable history of Bernstein’s
problem can be found in two classical surveys by Akhiezer [1]
and Mergelyan [13], a recent one by Lubinsky [12], or in the first volume
of Koosis’ book [8].
A simple if and only if solution in Bernstein’s problem most likely does
not exist. We plan to discuss some of the classical conditions along
with recent progress in these lectures.

ENTIRE FUNCTIONS AND COMPLETENESS PROBLEMS 3
The Beurling-Malliavin problem
Let Λ be a subset of the complex plane. Denote by EΛ the set of
complex exponential functions with ’frequencies’ from Λ:
EΛ = {exp(2πiλt)| λ ∈ Λ}.
The most standard example of such a set of functions is
EZ = {e 2πin }n∈Z,
which forms an orthonormal basis in L 2 ([0, 1]). A natural extension of
this important example is the following question.
Consider the case when Λ = {λn} is a general sequence of complex
numbers. Under what conditions on the frequencies Λ will the system
of exponentials EΛ be complete in L 2 (0, a)?
This natural question occupied analysts for many decades. The history
if its solution can be started from a theorem by Paley and Wiener
(1935) that says that if a real sequence Λ has upper density greater
than a, i.e.
lim sup
x→∞
then EΛ is complete in L2 (0, a).
#(Λ ∪ (0, x))
> a,
x
An immediate question is wether the statement of the theorem can
be reversed. Examples by Levinson, Kahane and Koosis showed that
no converse statement can be formulated using the primitive upper
density, utilized by Payley and Wiener.
A solution to that problem was obtained by Beurling and Malliavin
in a series of papers in the early 1960’s. Instead of the upper density,
they defined the so-called effective density of a sequence that allows one
to replace an inequality in the Payley-Wiener result with an equation
(after some additional reductions, that will be discussed in the future
lectures.)
To obtain the formula for the ’completeness radius’ of a sequence of
exponentials Beurling and Malliavin proved three intermediate results
that are now known as the first BM theorem, the little multiplier theorem
and the big multiplier theorem. Each of these results has independent
value and usage. Together these three theorems and the final
result, the second BM theorem, form the so-called Beurling-Malliavin
theory, that is considered to be one of the deepest parts of the 20’s
century Harmonic Analysis. Modern treatments of the BM theory can
be found in [7, 8] along with further references.

4 A. POLTORATSKI
The Type Problem
Consider a family Ea = E[0,a] of exponential functions whose frequencies
belong to the interval from 0 to a. If µ is a finite positive measure on
R we denote by Tµ its exponential type that is defined as
Tµ = inf{ a > 0 | Ea is complete in L 2 (µ) } (0.2)
if the set of such a is non-empty and as infinity otherwise. The type
problem asks to calculate Tµ in terms of µ.
This question first appears in the work of Wiener, Kolmogorov and
Krein in the context of stationary Gaussian processes that play an
important role in Probability Theory (see [9, 10] or the book by Dym
and McKean [5]). If µ is a spectral measure of a stationary Gaussian
process, completeness of Ea in L 2 (µ) is equivalent to the property that
the process at any time is determined by the data for the time period
from 0 to a. Hence the type of the measure is the minimal length of
the period of observation necessary to predict the rest of the process.
Since any even measure is a spectral measure of a stationary Gaussian
process, and vice versa, this reformulation is practically equivalent.
Important connections with spectral theory of second order differential
operators were studied by Gelfand and Levitan [6] and Krein [10, 11].
For more on the history and connections of the type problem see, for
instance, a note by Dym [4] or a recent paper by Borichev and Sodin
[3].

ENTIRE FUNCTIONS AND COMPLETENESS PROBLEMS 5
Questions and Exercises
Let Λ = {λn} be a real sequence. Define its radius of completeness as
Main example:
R(Λ) = sup{a | EΛ is complete in L 2 (0, a)}.
R(Z) = 1
(since EZ is an orthonormal basis of L 2 (0, 1)). Now let us consider one
half of Z, the set 2Z of all even integers. Show that R(2Z) is 1/2.
More generally, if we take 1/n-th of Z, nZ, then the radius decreases
correspondingly: R(nZ) = 1
n .
Question: Let us take one-half of Z in a different way and see what
happens. Find R(N), the radius of completeness of the set of positive
integers (use the Payley-Wiener theorem; the answer may surprise you).
Question: May R(Λ) change if one deletes finitely many points from
Λ? Give an example of an infinite sequence A ⊂ Z such that R(Z\A) =
R(Z) = 1.
Question: Can A from the previous example have positive upper density?
Question: Does a complete set have to contain a basis? A basis in
a Banach space is a set B of vectors such that every vector in the
space can be uniquely represented as a series of constant multiples of
vectors from B, converging in the norm. (Hint: consider, for instance,
the set of monomials in C([0, 1]). Try to describe the set of functions
representable as uniformly convergent power series.)

6 A. POLTORATSKI
References
[1] N. I. Akhiezer, On the weighted approximation of continuous functions by
polynomials on the real axis, Uspekhi Mat. Nauk 11(56), 3-43, AMS Transl.
(ser 2), 22 (1962), 95-137.
[2] S. N. Bernstein, Le probleme de lapproximation des fonctions continues sur
tout laxe reel et lune de ses applications, Bull. Math. Soc. France, 52(1924),
399410.
[3] Borichev, A., Sodin, M.Weighted exponential approximation and nonclassical
orthogonal spectral measures, preprint, arXiv:1004.1795v1
[4] Dym, H. On the span of trigonometric sums in weighted L 2 spaces, Linear and
Complex Analysis Problem Book 3, Part II, Lecture Notes in Math., Springer,
1994, 87 – 88
[5] Dym H, McKean H.P. Gaussian processes, function theory and the inverse
spectral problem Academic Press, New York, 1976
[6] Gelfand, I, M., Levitan, B. M. On the determination of a differential
equation from its spectral function (Russian), Izvestiya Akad. Nauk SSSR, Ser.
Mat., 15 (1951), 309–360; English translation in Amer. Math. Soc. Translation,
Ser. 2, 1 (1955), 253–304.
[7] Havin, V., Jöricke, B. The uncertainty principle in harmonic analysis.
Springer-Verlag, Berlin, 1994.
[8] Koosis, P. The logarithmic integral, Vol. I& II, Cambridge Univ. Press, Cambridge,
1988
[9] Krein, M. G. On an extrapolation problem of A. N. Kolmogorov, Dokl. Akad.
Nauk SSSR 46 (1945), 306–309 (Russian).
[10] Krein, M. G. On a basic approximation problem of the theory of extrapolation
and filtration of stationary random processes, Doklady Akad. Nauk SSSR
(N.S.) 94, (1954), 13–16 (Russian).
[11] Krein, M. G. On the transfer function of a one- dimensional boundary problem
of the second order (Russian), Doklady Akad. Nauk SSSR (N.S.) 88 (1953),
405–408.
[12] D. S. Lubinsky A Survey of Weighted Polynomial Approximation with Exponential
Weights, Surveys in Approximation Theory, 3, 1105 (2007)
[13] S. Mergelyan, Weighted approximation by polynomials, Uspekhi mat. nauk,
11 (1956), 107-152, English translation in Amer. Math. Soc. Translations, Ser
2, 10 (1958), 59-106